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NASA NTRS2010

Capacity Maximizing Constellations

Some non-traditional signal constellations have been proposed for transmission of data over the Additive White Gaussian Noise (AWGN) channel using such channel-capacity-approaching codes as low-density parity-check (LDPC) or turbo codes. Computational simulations have shown performance gains of more than 1 dB over traditional constellations. These gains could be translated to bandwidth- efficient communications, variously, over longer distances, using less power, or using smaller antennas. The proposed constellations have been used in a bit-interleaved coded modulation system employing state-ofthe-art LDPC codes. In computational simulations, these constellations were shown to afford performance gains over traditional constellations as predicted by the gap between the parallel decoding capacity of the constellations and the Gaussian capacity

Barsoum, Maged↗

DOE OSTI2025

Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

Decoding quantum error-correcting codes is a key challenge in enabling fault-tolerant quantum computation. In the classical setting, linear programming (LP) decoders offer provable performance guarantees and can leverage fast practical optimization algorithms. Although LP decoders have been proposed for quantum codes, their performance and limitations remain relatively underexplored. In this work, we uncover a key limitation of LP decoding for quantum low-density parity-check (LDPC) codes: certain constant-weight error patterns lead to ambiguous fractional solutions that cannot be resolved through independent rounding. To address this issue, we incorporate a post-processing technique known as ordered statistics decoding (OSD), which significantly enhances LP decoding performance in practice. Our results show that LP decoding, when augmented with OSD, can outperform belief propagation with the same post-processing for intermediate code sizes of up to hundreds of qubits. These findings suggest that LP-based decoders, equipped with effective post-processing, offer a promising approach for decoding near-term quantum LDPC codes.

Gu, Shouzhen [Yale U.]↗

DOE OSTI2025

Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

Gu, Shouzhen [Yale U.]↗

DOE OSTI2025

Power and Limitations of Linear Programming Decoder for Quantum LDPC Codes

Gu, Shouzhen [Yale U.]↗

NASA NTRS2010

Performance of Low-Density Parity-Check Coded Modulation

This paper reports the simulated performance of each of the nine accumulate-repeat-4-jagged-accumulate (AR4JA) low-density parity-check (LDPC) codes [3] when used in conjunction with binary phase-shift-keying (BPSK), quadrature PSK (QPSK), 8-PSK, 16-ary amplitude PSK (16- APSK), and 32-APSK.We also report the performance under various mappings of bits to modulation symbols, 16-APSK and 32-APSK ring scalings, log-likelihood ratio (LLR) approximations, and decoder variations. One of the simple and well-performing LLR approximations can be expressed in a general equation that applies to all of the modulation types.

Hamkins, Jon↗

NASA NTRS2010

Optimal Codes for the Burst Erasure Channel

Deep space communications over noisy channels lead to certain packets that are not decodable. These packets leave gaps, or bursts of erasures, in the data stream. Burst erasure correcting codes overcome this problem. These are forward erasure correcting codes that allow one to recover the missing gaps of data. Much of the recent work on this topic concentrated on Low-Density Parity-Check (LDPC) codes. These are more complicated to encode and decode than Single Parity Check (SPC) codes or Reed-Solomon (RS) codes, and so far have not been able to achieve the theoretical limit for burst erasure protection. A block interleaved maximum distance separable (MDS) code (e.g., an SPC or RS code) offers near-optimal burst erasure protection, in the sense that no other scheme of equal total transmission length and code rate could improve the guaranteed correctible burst erasure length by more than one symbol. The optimality does not depend on the length of the code, i.e., a short MDS code block interleaved to a given length would perform as well as a longer MDS code interleaved to the same overall length. As a result, this approach offers lower decoding complexity with better burst erasure protection compared to other recent designs for the burst erasure channel (e.g., LDPC codes). A limitation of the design is its lack of robustness to channels that have impairments other than burst erasures (e.g., additive white Gaussian noise), making its application best suited for correcting data erasures in layers above the physical layer. The efficiency of a burst erasure code is the length of its burst erasure correction capability divided by the theoretical upper limit on this length. The inefficiency is one minus the efficiency. The illustration compares the inefficiency of interleaved RS codes to Quasi-Cyclic (QC) LDPC codes, Euclidean Geometry (EG) LDPC codes, extended Irregular Repeat Accumulate (eIRA) codes, array codes, and random LDPC codes previously proposed for burst erasure protection. As can be seen, the simple interleaved RS codes have substantially lower inefficiency over a wide range of transmission lengths.

Hamkins, Jon↗

NASA NTRS2011

Frame Synchronization Without Attached Sync Markers

We describe a method to synchronize codeword frames without making use of attached synchronization markers (ASMs). Instead, the synchronizer identifies the code structure present in the received symbols, by operating the decoder for a handful of iterations at each possible symbol offset and forming an appropriate metric. This method is computationally more complex and doesn't perform as well as frame synchronizers that utilize an ASM; nevertheless, the new synchronizer acquires frame synchronization in about two seconds when using a 600 kbps software decoder, and would take about 15 milliseconds on prototype hardware. It also eliminates the need for the ASMs, which is an attractive feature for short uplink codes whose coding gain would be diminished by the overheard of ASM bits. The lack of ASMs also would simplify clock distribution for the AR4JA low-density parity-check (LDPC) codes and adds a small amount to the coding gain as well (up to 0.2 dB).

Hamkins, Jon↗

DOE OSTI2025

Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations

We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians 𝐻 0 defined by ⟦𝑁,𝐾,𝑑⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance 𝑑 ≥ 𝑐⁢log (𝑁), implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by 𝐻 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the 𝑂⁡(1) smallest eigenvalues of 𝐻 0 . The band originating from the smallest eigenvalue has 2 𝐾 states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth 𝛿 =𝐶⁢𝑁⁢𝑒 −Θ⁡(𝑑) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.

mathematical physics↗

DOE OSTI2025

Low-Density Parity-Check Codes as Stable Phases of Quantum Matter

Phases of matter with robust ground-state degeneracy, such as the quantum toric code, are known to be capable of robust quantum information storage. Here, we address the converse question: given a quantum error-correcting code, when does it define a stable gapped quantum phase of matter, whose ground-state degeneracy is robust against perturbations in the thermodynamic limit? We prove that a low-density parity-check (LDPC) code defines such a phase, robust against all few-body perturbations, if its code distance grows at least logarithmically in the number of degrees of freedom, and it exhibits “check soundness.” Many constant-rate quantum LDPC expander codes have such properties, and define stable phases of matter with a constant zero-temperature entropy density, violating the third law of thermodynamics. Our results also show that quantum toric-code phases are robust to spatially nonlocal few-body perturbations. Similarly, phases of matter defined by classical codes are stable against symmetric perturbations. In the classical setting, we present improved locality bounds on the quasiadiabatic evolution operator between two nearby states in the same code phase.

quantum error correction↗

NASA NTRS2007

Discussion on LDPC Codes and Uplink Coding

This slide presentation reviews the progress that the workgroup on Low-Density Parity-Check (LDPC) for space link coding. The workgroup is tasked with developing and recommending new error correcting codes for near-Earth, Lunar, and deep space applications. Included in the presentation is a summary of the technical progress of the workgroup. Charts that show the LDPC decoder sensitivity to symbol scaling errors are reviewed, as well as a chart showing the performance of several frame synchronizer algorithms compared to that of some good codes and LDPC decoder tests at ESTL. Also reviewed is a study on Coding, Modulation, and Link Protocol (CMLP), and the recommended codes. A design for the Pseudo-Randomizer with LDPC Decoder and CRC is also reviewed. A chart that summarizes the three proposed coding systems is also presented.

uplink↗

NASA NTRS2009

Joint Carrier-Phase Synchronization and LDPC Decoding

A method has been proposed to increase the degree of synchronization of a radio receiver with the phase of a suppressed carrier signal modulated with a binary- phase-shift-keying (BPSK) or quaternary- phase-shift-keying (QPSK) signal representing a low-density parity-check (LDPC) code. This method is an extended version of the method described in Using LDPC Code Constraints to Aid Recovery of Symbol Timing (NPO-43112), NASA Tech Briefs, Vol. 32, No. 10 (October 2008), page 54. Both methods and the receiver architectures in which they would be implemented belong to a class of timing- recovery methods and corresponding receiver architectures characterized as pilotless in that they do not require transmission and reception of pilot signals. The proposed method calls for the use of what is known in the art as soft decision feedback to remove the modulation from a replica of the incoming signal prior to feeding this replica to a phase-locked loop (PLL) or other carrier-tracking stage in the receiver. Soft decision feedback refers to suitably processed versions of intermediate results of iterative computations involved in the LDPC decoding process. Unlike a related prior method in which hard decision feedback (the final sequence of decoded symbols) is used to remove the modulation, the proposed method does not require estimation of the decoder error probability. In a basic digital implementation of the proposed method, the incoming signal (having carrier phase theta theta (sub c) plus noise would first be converted to inphase (I) and quadrature (Q) baseband signals by mixing it with I and Q signals at the carrier frequency [wc/(2 pi)] generated by a local oscillator. The resulting demodulated signals would be processed through one-symbol-period integrate and- dump filters, the outputs of which would be sampled and held, then multiplied by a soft-decision version of the baseband modulated signal. The resulting I and Q products consist of terms proportional to the cosine and sine of the carrier phase cc as well as correlated noise components. These products would be fed as inputs to a digital PLL that would include a number-controlled oscillator (NCO), which provides an estimate of the carrier phase, theta(sub c).

Simon, Marvin↗

NASA NTRS2010

Landsat Data Continuity Mission (LDCM) - Optimizing X-Band Usage

The NASA version of the low-density parity check (LDPC) 7/8-rate code, shortened to the dimensions of (8160, 7136), has been implemented as the forward error correction (FEC) schema for the Landsat Data Continuity Mission (LDCM). This is the first flight application of this code. In order to place a 440 Msps link within the 375 MHz wide X band we found it necessary to heavily bandpass filter the satellite transmitter output . Despite the significant amplitude and phase distortions that accompanied the spectral truncation, the mission required BER is maintained at < 10(exp -12) with less than 2 dB of implementation loss. We utilized a band-pass filter designed ostensibly to replicate the link distortions to demonstrate link design viability. The same filter was then used to optimize the adaptive equalizer in the receiver employed at the terminus of the downlink. The excellent results we obtained could be directly attributed to the implementation of the LDPC code and the amplitude and phase compensation provided in the receiver. Similar results were obtained with receivers from several vendors.

Garon, H. M.↗

NASA NTRS2008

Integrated Performance of Next Generation High Data Rate Receiver and AR4JA LDPC Codec for Space Communications

Low-density parity-check (LDPC) codes are the state-of-the-art in forward error correction (FEC) technology that exhibits capacity approaching performance. The Jet Propulsion Laboratory (JPL) has designed a family of LDPC codes that are similar in structure and therefore, leads to a single decoder implementation. The Accumulate-Repeat-by-4-Jagged- Accumulate (AR4JA) code design offers a family of codes with rates 1/2, 2/3, 4/5 and lengths 1024, 4096, 16384 information bits. Performance is less than one dB from capacity for all combinations.Integrating a stand-alone LDPC decoder with a commercial-off-the-shelf (COTS) receiver faces additional challenges than building a single receiver-decoder unit from scratch. In this work, we outline the issues and show that these additional challenges can be over-come by simple solutions. To demonstrate that an LDPC decoder can be made to work seamlessly with a COTS receiver, we interface an AR4JA LDPC decoder developed on a field-programmable gate array (FPGA) with a modern high data rate receiver and mea- sure the combined receiver-decoder performance. Through optimizations that include an improved frame synchronizer and different soft-symbol scaling algorithms, we show that a combined implementation loss of less than one dB is possible and therefore, most of the coding gain evidence in theory can also be obtained in practice. Our techniques can benefit any modem that utilizes an advanced FEC code.

Cheng, Michael K.↗

NASA NTRS2009

LDPC-PPM Coding Scheme for Optical Communication

In a proposed coding-and-modulation/demodulation-and-decoding scheme for a free-space optical communication system, an error-correcting code of the low-density parity-check (LDPC) type would be concatenated with a modulation code that consists of a mapping of bits to pulse-position-modulation (PPM) symbols. Hence, the scheme is denoted LDPC-PPM. This scheme could be considered a competitor of a related prior scheme in which an outer convolutional error-correcting code is concatenated with an interleaving operation, a bit-accumulation operation, and a PPM inner code. Both the prior and present schemes can be characterized as serially concatenated pulse-position modulation (SCPPM) coding schemes. Figure 1 represents a free-space optical communication system based on either the present LDPC-PPM scheme or the prior SCPPM scheme. At the transmitting terminal, the original data (u) are processed by an encoder into blocks of bits (a), and the encoded data are mapped to PPM of an optical signal (c). For the purpose of design and analysis, the optical channel in which the PPM signal propagates is modeled as a Poisson point process. At the receiving terminal, the arriving optical signal (y) is demodulated to obtain an estimate (a^) of the coded data, which is then processed by a decoder to obtain an estimate (u^) of the original data.

Barsoum, Maged↗

NASA NTRS2013

Bilayer Protograph Codes for Half-Duplex Relay Channels

Direct to Earth return links are limited by the size and power of lander devices. A standard alternative is provided by a two-hops return link: a proximity link (from lander to orbiter relay) and a deep-space link (from orbiter relay to Earth). Although direct to Earth return links are limited by the size and power of lander devices, using an additional link and a proposed coding for relay channels, one can obtain a more reliable signal. Although significant progress has been made in the relay coding problem, existing codes must be painstakingly optimized to match to a single set of channel conditions, many of them do not offer easy encoding, and most of them do not have structured design. A high-performing LDPC (low-density parity-check) code for the relay channel addresses simultaneously two important issues: a code structure that allows low encoding complexity, and a flexible rate-compatible code that allows matching to various channel conditions. Most of the previous high-performance LDPC codes for the relay channel are tightly optimized for a given channel quality, and are not easily adapted without extensive re-optimization for various channel conditions. This code for the relay channel combines structured design and easy encoding with rate compatibility to allow adaptation to the three links involved in the relay channel, and furthermore offers very good performance. The proposed code is constructed by synthesizing a bilayer structure with a pro to graph. In addition to the contribution to relay encoding, an improved family of protograph codes was produced for the point-to-point AWGN (additive white Gaussian noise) channel whose high-rate members enjoy thresholds that are within 0.07 dB of capacity. These LDPC relay codes address three important issues in an integrative manner: low encoding complexity, modular structure allowing for easy design, and rate compatibility so that the code can be easily matched to a variety of channel conditions without extensive re-optimization. The main problem of half-duplex relay coding can be reduced to the simultaneous design of two codes at two rates and two SNRs (signal-to-noise ratios), such that one is a subset of the other. This problem can be addressed by forceful optimization, but a clever method of addressing this problem is via the bilayer lengthened (BL) LDPC structure. This method uses a bilayer Tanner graph to make the two codes while using a concept of "parity forwarding" with subsequent successive decoding that removes the need to directly address the issue of uneven SNRs among the symbols of a given codeword. This method is attractive in that it addresses some of the main issues in the design of relay codes, but it does not by itself give rise to highly structured codes with simple encoding, nor does it give rate-compatible codes. The main contribution of this work is to construct a class of codes that simultaneously possess a bilayer parity- forwarding mechanism, while also benefiting from the properties of protograph codes having an easy encoding, a modular design, and being a rate-compatible code.

Divsalar, Dariush↗

NASA NTRSDate not supplied

NASA's Deep Space Network (DSN) Lunar Exploration Upgrades (DLEU)

In the near future, the National Aeronautics and Space Administration (NASA) will return to the moon beginning the next era of human exploration. NASA’s Space Communications and Navigation (SCaN) program will play a vital role in establishing communications and navigation support to realize the ambitious goals of the Artemis program. SCaN’s overall lunar communications support plan will be covered in a separate 2023 SpaceOps paper: “NASA’s Communications and Navigation Architecture Plans to Support the Return to the Moon and a Sustainable Lunar Presence”. The plan as it currently stands, includes a three-fold approach of lunar relay services, a dedicated set of new ground stations and support through the Deep Space Network (DSN). This paper will have a more granular focus on the DSN and NASA’s plans to upgrade and expand the network to be better suited for human spaceflight on and around the lunar surface. NASA’s Deep Space Network (DSN) will be a critical communications component for the upcoming lunar activities. There will be multiple spacecraft, using different bands, and some of those spacecraft will be transmitting and receiving using multiple bands, requiring DSN support of S-band (2 GHz), X-band (7 GHz up, 8 GHz down), and K-band (22.5 GHz up, 26 GHz down). Since there may be more than one spacecraft in the beamwidth of the DSN antennas, the DSN support will require an extension of the DSN’s capability to support multiple spacecraft using one antenna, expanding it to provide two simultaneous uplinks in the different bands at each antenna. Achieving this requires using new techniques for manufacturing the frequency selective surfaces, called dichroics, which steer the different frequency beams from and to the appropriate transmitting and receiving equipment, along with the addition of a new K-band uplink system. Additionally, due to the relative closeness of the moon from Earth (as opposed to the planetary missions the DSN supports daily), significantly higher data rates on both uplink and downlink are required, specifically up to 20 Mbps on the uplink and 150 Mbps on the downlink, both using Low Density Parity Check (LDPC) error correcting codes. And, again due to the relative closeness of the moon, there is a need for low latency data delivery of the high rate downlink telemetry which requires a change in the current DSN paradigm of delivering higher rate data with higher latency.

deep space network↗

NASA NTRSDate not supplied

NASA’s Deep Space Network (DSN) Lunar Exploration Upgrades (DLEU)

In the near future, the National Aeronautics and Space Administration (NASA) will returnhumansto the moon beginning the next era of human exploration. NASA’s Space Communications and Navigation (SCaN) program will play a vital role in establishing communications and navigation support to realize the ambitious goals of the Artemis program. SCaN’s overall lunar communications support plan will be covered in a separate 2023 SpaceOps paper: “NASA’s Communications and Navigation Architecture Plans to Support the Return to the Moon and a Sustainable Lunar Presence.” The four-point plan,as it currently stands, includes lunar relay services, a dedicated set of new ground stations, international partner contributions,and supportthrough the Deep Space Network (DSN)and associated upgrades. This paper will have a more granular focus on the DSN and NASA’s plans to upgrade and expand the network to be better suited for human spaceflight on and around the lunar surface. NASA’s Deep Space Network (DSN) will be a critical communications component for the upcoming lunar activities. There will be multiple spacecraft, using different bands, and some of those spacecraft will be transmitting and receiving using multiple bands, requiring DSN support of S-band (2 GHz), X-band (7 GHz up, 8 GHz down), and K-band (22.5 GHz up, 26 GHz down). Since there may be more than one spacecraft in the beamwidth of the DSN antennas, the DSN support will require an extension of the DSN’s capability to support multiple spacecraft using one antenna, expanding it to provide two simultaneous uplinks in the different bands at each antenna. Achieving this requires using new techniques for manufacturing the frequency selective surfaces, called dichroics, which steer the different frequency beams from and to the appropriate transmitting and receiving equipment, along with the addition of a new K-band uplink system. Additionally, due to the relative closeness of the moon from Earth (as opposed to the planetary missionsthe DSN supports daily), significantly higher data rates on both uplink and downlinkare requiredare possible and desirable by the lunar missions, specifically up to 20 Mbps on the uplink and 150 Mbps on the downlink, both using Low Density Parity Check (LDPC) error correcting codes. And, again due to the relative closeness of the moon, there is a need for low latency data delivery of the high rate downlink telemetry which requires a change in the current DSN paradigm of delivering higher rate data with higher latency.

Moon↗

NASA NTRS2019

Bit Error Rate and Frame Error Rate Data Processing for Space Communications and Navigation-Related Communication System Analysis Tools

One of the capabilities that the Space Communications and Navigation (SCaN) Strategic Center for Networking, Integration, and Communications (SCENIC) user interface (UI) web application intends to provide its users is the addition of network protocol and link encryption augmentations of communication system analyses. Before any of these analyses capabilities can be modeled, the simulations of bit error rate (BER) and frame error rate (FER) against signal-to-noise ratio (SNR) have been conducted, requiring parameters from several known coding types (low-density parity-check (LDPC), convolutional, etc.), signal modulations (binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), etc.), coding rates (1/2, 1/3, etc.), and frame sizes (1,280, 3,580, etc.). However, in order to extract useful information from the results of these simulations, a curve fitting technique has been applied to each resulting dataset to extend and extrapolate the curve fit of BER and FER down to 10–30 using MATLAB® Curve Fitting Toolbox™ (The MathWorks, Inc.). This is a necessary step because simulations of BER and FER were only performed to around 10–9 due to the extensive simulation time that would be required to obtain significant simulation results at the error levels desired. Furthermore, the fitted curve results were applied to a finer resolution for the SNR at 0.01-dB interval instead of the 0.05-dB interval limitation used in the simulation. All the possible combinations of the coding types, signal modulations, coding rates, frame sizes, and the extension of BER and FER curves would enable users to capture a wide range of link performances that directly relates to the addition of higher level networking data encapsulated in a frame. The curve fitting results also led to the modeling of the optical link error rate performance by solving for coding gain, FER_BER SNR delta, coded optical BER-SNR, and coded optical FER–SNR.

Communication link analysis↗